There is a common result that if a function $f$ is locally integrable then there is a condition for the distribution derivative $f'$ to be a complex measure. One of them is that $f$ agrees a.e. with a function of bounded variation $F$. And if so, we have $\langle f',\phi\rangle=\int\varphi dF$. Was looking for a reference for this proof. (And why this fails as a necessary condition)
EDIT: Thanks to @reuns, it is easy to see why it is not necessary. Does strengthening to bounded variation with $\lim_{|x|\to\infty} f(x)=0$ do the trick?